The Gamma function

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The integrals in Gradshteyn and Ryzhik. Part 4:

The Gamma function

Article · June 2007

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Victor H. Moll Tulane University

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arXiv:0705.0179v1 [math.CA] 1 May 2007

THE INTEGRALS IN GRADSHTEYN AND RYZHIK. PART 4:

THE GAMMA FUNCTION.

VICTOR H. MOLL

Abstract. We present a systematic derivation of some definite integrals in the classical table of Gradshteyn and Ryzhik that can be reduced to the gamma function.

1. Introduction

The table of integrals [2] contains some evaluations that can be derived by elementary means from thegamma function, defined by

(1.1) Γ(a) =

Z ∞ 0

x^a⁻¹e⁻^xdx.

The convergence of the integral in (1.1) requires a > 0. The goal of this paper is to present some of these evaluations in a systematic manner. The techniques developed here will be employed in future publications. The reader will find in [1]

analytic information about this important function.

The gamma function represents the extension of factorials to real parameters.

The value

(1.2) Γ(n) = (n−1)!, forn∈N

is elementary. On the other hand, the special value

(1.3) Γ ¹₂

=√ π is equivalent to the well-knownnormal integral (1.4)

Z ∞ 0

exp(−t²)dt=¹₂Γ ¹₂ .

The reader will find in [1] proofs of Legendre’s duplication formula

(1.5) Γ x+¹₂

= Γ(2x)√ π Γ(x) 2^2x⁻¹, that produces forx=m∈Nthe values

(1.6) Γ m+¹₂

=

√π 2^2m

(2m)!

m! . This appears as3.371in [2].

Date: February 1, 2008.

1991Mathematics Subject Classification. Primary 33.

Key words and phrases. Integrals.

1

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2 VICTOR H. MOLL

2. The introduction of a parameter

The presence of a parameter in a definite integral provides great amount of flexibility. The change of variablesx=µtin (1.1) yields

(2.1) Γ(a) =µ^a

Z _∞

0

t^a⁻¹e⁻^µtdt.

This appears as3.381.4in [2] and the choicea=n+ 1, withn∈N, that reads (2.2)

Z ∞ 0

tⁿe⁻^µtdt=n!µ⁻ⁿ⁻¹ appears as3.351.3.

The special casea=m+¹₂, that appears as3.371in [2], yields (2.3)

Z _∞

0

t^m⁻¹²e⁻^µtdt=

√π 2^2m

(2m)!

m! µ⁻^m⁻¹², is consistent with (1.6).

The combination (2.4)

Z ∞

0

e⁻^νx−e⁻^µx

x^ρ+1 dx=µ^ρ−ν^ρ

ρ Γ(1−ρ),

that appears as3.434.1in [2] can now be evaluated directly. The parameters are restricted by convergence: µ, ν >0 andρ <1. The integral3.434.2

(2.5)

Z ∞

0

e⁻^µx−e⁻^νx

x dx= lnν µ,

is obtained from (2.4) by passing to the limit as ρ → 0. This is an example of Frullani integralsthat will be discussed in a future publication.

The reader will be able to check3.478.1:

(2.6)

Z ∞ 0

x^ν⁻¹exp(−µx^p)dx= 1 pµ⁻^ν/pΓ

ν p

, and3.478.2:

(2.7)

Z _∞

0

x^ν⁻¹[1−exp(−µx^p)]dx=−1

|p|µ⁻^ν/pΓ ν

p

by introducing appropriate parameter reduction.

The parameters can be used to prove many of the classical identities for Γ(a).

Proposition 2.1. The gamma function satisfies

(2.8) Γ(a+ 1) =aΓ(a).

Proof. Differentiate (2.1) with respect toµto produce (2.9) 0 =aµ^a⁻¹

Z _∞

0

t^a⁻¹e⁻^µtdt−µ^a Z _∞

0

t^ae⁻^µtdt.

Now putµ= 1 to obtain the result.

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Differentiating (1.1) with respect to the parameterayields

(2.10) Γ^′(a) =

Z _∞

0

x^a⁻¹e⁻^xlnx dx.

Further differentiation introduces higher powers of lnx:

(2.11) Γ⁽ⁿ⁾(a) =

Z ∞

0

x^a⁻¹e⁻^x (lnx)ⁿ dx.

In particular, fora= 1, we obtain:

(2.12)

Z ∞ 0

(lnx)ⁿe⁻^xdx= Γ⁽ⁿ⁾(1).

The special casen= 1 yields (2.13)

Z ∞

0

e⁻^xlnx dx= Γ^′(1).

The reader will find in [1], page 176 an elementary proof that Γ^′(1) =−γ, where

(2.14) γ:= lim

n→∞

n

X

k=1

1 k−lnn

is Euler’s constant. This is one of the fundamental numbers of Analysis.

On the other hand, differentiating (2.1) produces (2.15)

Z _∞

0

x^a⁻¹e⁻^µx(lnx)ⁿ dx= ∂

∂a ⁿ

µ⁻^aΓ(a) ,

that appears as4.358.5in [2]. Using Leibnitz’s differentiation formula we obtain (2.16)

Z ∞ 0

x^a⁻¹e⁻^µx(lnx)ⁿ dx=µ⁻^a

n

X

k=0

(−1)^k n

k

(lnµ)^kΓ⁽ⁿ⁻^k)(a).

In the special casea= 1 we obtain (2.17)

Z ∞ 0

e⁻^µx(lnx)ⁿ dx= 1 µ

n

X

k=0

(−1)^k n

k

(lnµ)^kΓ⁽ⁿ⁻^k)(1).

The casesn= 1,2,3 appear as 4.331.1,4.335.1and4.335.3respectively.

In order to obtain analytic expressions for the terms Γ⁽ⁿ⁾(1), it is convenient to introduce thepolygamma function

(2.18) ψ(x) = d

dxln Γ(x).

The derivatives ofψsatisfy

(2.19) ψ⁽ⁿ⁾(x) = (−1)ⁿ⁺¹n!ζ(n+ 1, x), where

(2.20) ζ(z, q) =

X∞

n=0

1 (n+q)^z is theHurwitz zeta function. In particular this gives (2.21) ψ⁽ⁿ⁾(1) = (−1)ⁿ⁺¹n!ζ(n+ 1).

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4 VICTOR H. MOLL

The values of Γ⁽ⁿ⁾(1) can now be computed by recurrence via

(2.22) Γ⁽ⁿ⁺¹⁾(1) =

n

X

k=0

n k

Γ^(k)(1)ψ⁽ⁿ⁻^k)(1), obtained by differentiating Γ^′(x) =ψ(x)Γ(x).

Using (2.19) the reader will be able to check the first few cases of (2.15), we employ the notationδ=ψ(a)−lnµ:

Z _∞

0

x^a⁻¹e⁻^µxln²x dx = Γ(a) µ^a

δ²+ζ(2, a) , Z _∞

0

x^a⁻¹e⁻^µxln³x dx = Γ(a) µ^a

δ³+ 3ζ(2, a)δ−2ζ(3, a) , Z ∞

0

x^a⁻¹e⁻^µxln⁴x dx = Γ(a) µ^a

δ⁴+ 6ζ(2, a)δ²−8ζ(3, a)δ+ 3ζ²(2, a) + 6ζ(4, a) . These appear as4.358.2,4.358.3and4.358.4, respectively.

3. Elementary changes of variables

The use of appropriate changes of variables yields, from the basic definition (1.1), the evaluation of more complicated definite integrals. For example, let x=t^b to obtain, withc=ab−1,

(3.1)

Z ∞ 0

t^cexp(−t^b)dt= 1 bΓ

c+ 1 b

. The special casea= 1/b, that is c= 0, is

(3.2)

Z ∞ 0

exp(−t^b)dt=1 bΓ

1 b

,

that appears as3.326.1in [2]. The special caseb= 2 is the normal integral (1.4).

We can now introduce an extra parameter viat=s^1/bx. This produces (3.3)

Z ∞

0

x^mexp(−sx^b)dx= Γ(a) s^ab ,

withm=ab−1. This formula appears (at least) three times in [2]: 3.326.2,3.462.9 and3.478.1. Moreover, the cases= 1, c= (m+ 1/2)n−1 and b=nappears as 3.473:

(3.4)

Z ∞

0

exp(−xⁿ)x

“ m+1

2

” n−1

dx= (2m−1)!!

2^mn

√π.

The form given here can be established using (1.6).

Differentiating (3.3) with respect to the parameter m (keeping in mind that a= (m+ 1)/b), yields

(3.5)

Z ∞

0

x^me⁻^sx^b lnx dx= Γ(a)

b²s^a [ψ(a)−lns]. In particular, ifb= 1 we obtain

(3.6)

Z _∞

0

x^me⁻^sx lnx dx=Γ(m+ 1)

s^m+1 [ψ(m+ 1)−lns].

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The casem= 0 andb= 2 gives (3.7)

Z ∞

0

e⁻^sx² lnx dx=−

√π 4√

s(γ+ ln 4s),

where we have usedψ(1/2) =−γ−2 ln 2. This appears as4.333in [2].

An interesting example isb=m= 2. Using the values

(3.8) Γ ³₂

=√

π/2 andψ ³₂

= 2−2 ln 2−γ the expression (3.5) yields

(3.9)

Z ∞

0

x²e⁻^sx² lnx dx= 1

8s(2−ln 4s−γ) rπ

s.

The values of ψ at half-integers follow directly from (1.5). Formula (3.9) appears as4.355.1in [2]. Using (3.5) it is easy to verify

(3.10)

Z ∞ 0

(µx²−n)x²ⁿ⁻¹e⁻^µx² lnx dx=(n−1)!

4µⁿ , and

(3.11)

Z _∞

0

(2µx²−2n−1)x²ⁿe⁻^µx² lnx dx=(2n−1)!!

2(2µ)ⁿ rπ

µ,

for n ∈N. These appear as, respectively, 4.355.3and 4.355.4 in [2]. The term (2n−1)!! is the semi-factorial defined by

(3.12) (2n−1)!! = (2n−1)(2n−3)· · ·5·3·1.

Finally, formula4.369.1in [2]

(3.13)

Z ∞

0

x^a⁻¹e⁻^µx[ψ(a)−lnx]dx=Γ(a) lnµ µ^a

can be established by the methods developed here. The more ambitious reader will check that

Z ∞

0

xⁿ⁻¹e⁻^µx

[lnx−¹2ψ(n)]²−¹2ψ^′(n) dx= (n−1)!

µⁿ

[lnµ−¹2ψ(n)]²+¹₂ψ^′(n) , that is4.369.2in [2].

We can also write (3.5) in the exponential scale to obtain (3.14)

Z _∞

−∞

te^mtexp −se^bt

dt=Γ(m/b) b²s^m/b

ψm b

−lns . The special caseb=m= 1 produces

(3.15)

Z ∞

−∞

te^texp −se^t

dt=−(γ+ lns) s

that appears as 3.481.1. The second special case, appearing as 3.481.2, is b = 2, m= 1, that yields

(3.16)

Z ∞

−∞

te^texp −se^2t dt=−

√π(γ+ ln 4s) 4√

s .

This uses the valueψ(1/2) =−(γ+ 2 ln 2).

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6 VICTOR H. MOLL

There are many other possible changes of variables that lead to interesting evaluations. We conclude this section with one more: letx=e^tto convert (1.1) into (3.17)

Z _∞

−∞

exp (−e^x)e^axdx= Γ(a).

This is3.328in [2].

As usual one should not prejudge the difficulty of a problem: the example3.471.3 states that

(3.18)

Z a

0

x⁻^µ⁻¹(a−x)^µ⁻¹e⁻^β/xdx=β⁻^µa^µ⁻¹Γ(µ) exp

−^βa

.

This can be reduced to the basic formula for the gamma function. Indeed, the change of variablest=β/xproduces

(3.19) I=β⁻^µa^µ⁻¹ Z ∞

β/a

(t−β/a)^µ⁻¹e⁻^tdt.

Now let y = t−β/a to complete the evaluation. The table [2] writes µ instead of a: it seems to be a bad idea to haveµ and uin the same formula, it leads to typographical errors that should be avoided.

Another simple change of variables gives the evaluation of3.324.2:

(3.20)

Z ∞

−∞

e⁻^(x⁻^b/x)²ⁿdx= 1 nΓ

1 2n

.

The symmetry yields

(3.21) I= 2

Z ∞ 0

e⁻^(x⁻^b/x)²ⁿdx.

The change of variablest=b/xyields, usingb >0,

(3.22) I= 2b

Z _∞

0

e⁻^(t⁻^b/t)²ⁿdt t².

The average of these forms produces

(3.23) I=

Z ∞ 0

e⁻^(x⁻^b/x)²ⁿ

1 + b x²

dx.

Finally, the change of variablesu=x−b/xgives the result. Indeed, letu=x−b/x and observe thatuis increasing whenb >0. This restriction is missing in the table.

Then we get

(3.24) I= 2

Z ∞ 0

e⁻^u²ⁿdu.

This can now be evaluated viav=u²ⁿ.

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Note. In the caseb <0 the change of variablesu=x−b/xhas an inverse with two branches, splitting atx=√

−b. Then we write I := 2

Z _∞

0

e⁻^(x⁻^b/x)²ⁿdx (3.25)

= 2

Z ^√−b 0

e⁻^(x⁻^b/x)²ⁿdx+ 2 Z ∞

√−b

e⁻^(x⁻^b/x)²ⁿdx.

The change of variablesu=x−b/xis now used in each of the integrals to produce

(3.26) I= 2

Z ∞ 2√

−b

uexp(−u²ⁿ)du

√u²+ 4b . The change of variablesz=√

u²+ 4b yields

(3.27) I= 2

Z ∞ 0

exp −(z²−4b)ⁿ . We are unable to simplify it any further.

4. The logarithmic scale Euler prefered the version

(4.1) Γ(a) =

Z 1

0

ln1

u a−1

du.

We will write this as

(4.2) Γ(a) =

Z 1

0

(−lnu)^a⁻¹ du,

for better spacing. Many of the evaluations in [2] follow this form. Section4.215 in [2] consists of four examples: the first one, 4.215.1is (4.1) itself. The second one, labeled4.215.2and written as

(4.3)

Z 1

0

dx

(−lnx)^µ = π

Γ(µ)cosecµπ, is evaluated as Γ(1−µ) by (4.1). The identity

(4.4) Γ(µ)Γ(1−µ) = π

sinπµ

yields the given form. The reader will find in [1] a proof of this identity. The section concludes with the special values

(4.5)

Z 1

0

√−lnx dx=

√π 2 , as4.215.3and4.215.4:

(4.6)

Z 1

0

√dx

−lnx=√ π.

Both of them are special cases of (4.1).

The reader should check the evaluations4.269.3:

(4.7)

Z 1

0

x^p⁻¹√

−lnx dx=1 2

rπ p³,

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8 VICTOR H. MOLL

and4.269.4:

(4.8)

Z 1

0

x^p⁻¹dx

√−lnx = rπ

p

by reducing them to (2.1). Also4.272.5, 4.272.6and4.272.7 Z ∞

1

(lnx)^pdx

x² = Γ(1 +p), (4.9)

Z 1

0

(−lnx)^µ⁻¹ x^ν⁻¹dx = 1 ν^µΓ(µ), Z 1

0

(−lnx)ⁿ⁻¹² x^ν⁻¹dx = (2n−1)!!

(2ν)ⁿ rπ

ν, can be evaluated directly in terms of the gamma function.

Differentiating (4.1) with respect toayields4.229.4in [2]:

(4.10)

Z 1

0

ln (−lnx) (−lnx)^a⁻¹dx= Γ^′(a) =ψ(a)Γ(a), withψ(a) defined in (2.18). The special casea= 1 is4.229.1:

(4.11)

Z 1

0

ln (−lnx)dx=−γ, and

(4.12)

Z 1

0

ln (−lnx) dx

√−lnx =−(γ+ 2 ln 2)√ π, that appears as4.229.3, is obtained by using the values Γ ¹₂

=√πandψ ¹₂

=

−(γ+ 2 ln 2).

The same type of arguments confirms4.325.11 (4.13)

Z 1

0

ln(−lnx)x^µ⁻¹dx

√−lnx =−(γ+ ln 4µ) rπ

µ, and4.325.12:

(4.14)

Z 1

0

ln(−lnx) (−lnx)^µ⁻¹ x^ν⁻¹dx= 1

ν^µΓ(µ) [ψ(µ)−lnν]. In particular, whenµ= 1 we obtain4.325.8:

(4.15)

Z 1

0

ln(−lnx)x^ν⁻¹dx=−1

ν(γ+ lnν). 5. The presence of fake parameters

There are many formulas in [2] that contain parameters. For example,3.461.2 states that

(5.1)

Z ∞ 0

x²ⁿe⁻^px²dx=(2n−1)!!

2(2p)ⁿ rπ

p and3.461.3states that

(5.2)

Z _∞

0

x²ⁿ⁺¹e⁻^px²dx= n!

2pⁿ⁺¹.

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The change of variablest=px²eliminates thefakeparameterpand reduces3.461.2 to

(5.3)

Z _∞

0

tⁿ⁻¹²e⁻^tdt=(2n−1)!!

2ⁿ

√π and3.461.3to

(5.4)

Z ∞ 0

tⁿe⁻^tdt=n!.

These are now evaluated by identifying them with Γ(n+¹₂) and Γ(n+ 1), respectively.

A second way to introduce fake parameters is to shift the integral (2.1) via s=t+bto produce

(5.5)

Z _∞

b

(s−b)^a⁻¹e⁻^sµds=µ⁻^ae⁻^µbΓ(a).

This appears as3.382.2in [2].

There are many more integrals in [2] that can be reduced to the gamma function.

These will be reported in a future publication.

Acknowledgments. The author wishes to thank Luis Medina for a careful reading of an earlier version of the paper. The partial support of NSF-DMS 0409968 is also acknowledged.

References

[1] G. Boros and V. Moll.Irresistible Integrals. Cambridge University Press, New York, 1st edition, 2004.

[2] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Edited by A.

Jeffrey and D. Zwillinger. Academic Press, New York, 6th edition, 2000.

Department of Mathematics, Tulane University, New Orleans, LA 70118 E-mail address: [email protected]